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A Splitting Condition and Geometric Rates of Convergence to Equilibrium

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Stationary Processes and Discrete Parameter Markov Processes

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 293))

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Abstract

This chapter builds on the representation of Markov processes in terms of i.i.d. iterated maps by developing the so-called “splitting techniques”that capture the recurrence structure of certain iterated maps in a novel way.

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Notes

  1. 1.

    See Bhattacharya and Majumdar (2007), Bhattacharya and Waymire (2002) for this and related refinements.

  2. 2.

    For further results on the problem of delineating the structure of π b beginning with early results of Erdos (1937), see Peres et al. (1999).

  3. 3.

    See Gelman et al. (1995) for an early exposition. Also see Chib and Greenberg (1995).

  4. 4.

    The Corollary 19.8 also extends to closed sets \(S\subset {\mathbb R}^k\), with a coordinatewise partial order. For this, see Bhattacharya and Lee (1988). Also see Bhattacharya and Majumdar (2010a), and Chakroborty and Rao (1998).

  5. 5.

    See BCPT Theorem 7.1, pp. 137–139; Billingsley (1968): pp. 11–14.

  6. 6.

    [Personal communication (1994)] This example was kindly furnished by Professor B.V. Rao, ISI Kolkata, India.

  7. 7.

    See Bhattacharya and Lee (1997).

  8. 8.

    Some general insights and detailed analysis for subsets S of \({\mathbb R}\) and \({\mathbb R}^2\) may be found in Chakroborty and Rao (1998).

  9. 9.

    For the details, we refer to Bhattacharya and Majumdar (2007), p. 315.

  10. 10.

    [Personal Communication (2019)] This was kindly pointed out by Dr. Eduardo A. Silva of the Universidade de Brasilia, Brazil.

  11. 11.

    For the speed of convergence to the invariant probability in these examples, see Lund and Tweedie (1996) and Bhattacharya and Majumdar (2010a) (with polynomial rates).

  12. 12.

    Markov processes generated by i.i.d. iterations of quadratic maps were considered by Bhattacharya and Rao (1993). Also see Athreya and Dai (2000), Bhattacharya and Majumdar (2004), (2007).

  13. 13.

    These examples and much more are presented in Bhattacharya and Majumdar (2007).

  14. 14.

    This example may be found in Mirman (1980).

  15. 15.

    In the physics literature, both in the definition of H(s) and the exponent, defining π, each, has a minus sign. The signs are included there to make alignment of spins the least energetic (ground states), as well as to make them the most likely in the case β > 0. However, since they cancel, we omit them for convenience.

  16. 16.

    A version of this so-called coupling from the past method already appears in the proof of Proposition 19.7. The application to Monte Carlo simulations was effectively developed by Propp and Wilson (1998).

  17. 17.

    See Haggstrom and Nelander (1998) for an extension exploiting earlier ideas of Wilfrid Kendall.

References

  • Athreya KB, Dai J (2000) Random logistic maps I. J Theor Probab 13(2):595–608

    Article  MathSciNet  MATH  Google Scholar 

  • Bhattacharya R, Lee O (1988) Asymptotics of a class of Markov processes which are not in general irreducible. Ann Probab 16(3):1333–1347

    Article  MathSciNet  MATH  Google Scholar 

  • Bhattacharya R, Lee O (1997) Correction: asymptotics of a class of Markov processes which are not in general irreducible [Ann. Probab. 16(3):1333–1347] Ann Probab 25(3):1541–1543

    Article  MathSciNet  Google Scholar 

  • Bhattacharya R, Majumdar M (2004) Stability in distribution of randomly per-turbed quadratic maps as Markov processes. Ann Appl Probab, pp 1802–1809

    Google Scholar 

  • Bhattacharya R, Majumdar M (2007) Random dynamical systems: theory and applications. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Bhattacharya R, Majumdar M (2010a) Random iterates of monotone maps. Rev Econ Res 14:185–192

    MathSciNet  MATH  Google Scholar 

  • Bhattacharya R, Waymire E (2002) An approach to the existence of unique invariant probabilities for Markov processes. In: Limit theorems in probability and statistics, vol I. János Bolyai Math. Soc., pp 181–200

    Google Scholar 

  • Billingsley P (1968) Convergence of probability measures. Wiley, New York

    MATH  Google Scholar 

  • Chakroborty S, Rao BV (1998) Completeness of the Bhattacharya metric on the space of probabilities. Statist Probab Lett 36:321–326

    Article  MathSciNet  MATH  Google Scholar 

  • Chib S, Greenberg E (1995) Understanding the . Am Statist 49(4):327–335

    Google Scholar 

  • Dubins LE, Freedman DA (1966) Invariant probabilities for certain Markov processes. Ann Math Statist 37(4):837–848

    Article  MathSciNet  MATH  Google Scholar 

  • Erdos P (1937) Some problems and results in elementary number theory. Proc Camb Phil Soc 33:6–12

    Google Scholar 

  • Gelman A, Carlin JB, Stern HS, Rubin DB (1995) Bayesian data analysis. Chapman and Hall/CRC, London

    Book  MATH  Google Scholar 

  • Haggstrom O, Nelander K (1998) Exact sampling from anti-monotone systems. Statistica Neerlandica52:360

    Google Scholar 

  • Lund RB, Tweedie RL (1996) Geometric convergence rates for stochastically ordered Markov chains. Math Oper Res 21:182–196

    Article  MathSciNet  MATH  Google Scholar 

  • Mirman LJ (1980) One sector economic growth and uncertainty: a survey. Stoch Program 537–567

    Google Scholar 

  • Peres Y, Shlag W, Solomyak B (1999) Absolute continuity of Bernoulli convolutions, a simple proof. Math Res Lett 3:231–239

    Article  MathSciNet  MATH  Google Scholar 

  • Propp J, Wilson D (1998) Coupling from the past: a user’s guide. Microsurveys in discrete probability (Princeton, NJ, 1997), DIMACS Ser.Discrete Math. Theoret. Comput. Sci., vol 41. American Mathematical Society, Providence, pp 181–192

    Google Scholar 

  • Solomyak B (1995) On the random series \(\sum \pm \lambda ^n\) (an Erdos problem). Ann Math 142:611–625

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

    Rabi Bhattacharya

  2. Department of Mathematics, Oregon State Univeristy, Corvallis, OR, USA

    Edward Waymire

Authors
  1. Rabi Bhattacharya
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  2. Edward Waymire
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Bhattacharya, R., Waymire, E. (2022). A Splitting Condition and Geometric Rates of Convergence to Equilibrium. In: Stationary Processes and Discrete Parameter Markov Processes. Graduate Texts in Mathematics, vol 293. Springer, Cham. https://doi.org/10.1007/978-3-031-00943-3_19

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